# Rings and Integral Domains

• Sep 16th 2008, 09:13 AM
Aryth
Rings and Integral Domains
Let $\displaystyle \mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\displaystyle \mathcal{Z}$ and the usual operations of addition and multiplication.

Let $\displaystyle S$ be the set of polynomials in $\displaystyle \mathcal{Z}[x]$ with roots at $\displaystyle x=0$ and $\displaystyle x=1$. Prove that $\displaystyle S$ is a subring of $\displaystyle \mathcal{Z}[x]$.
• Sep 16th 2008, 09:37 AM
ThePerfectHacker
Quote:

Originally Posted by Aryth
Let $\displaystyle \mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\displaystyle \mathcal{Z}$ and the usual operations of addition and multiplication.

Let $\displaystyle S$ be the set of polynomials in $\displaystyle \mathcal{Z}[x]$ with roots at $\displaystyle x=0$ and $\displaystyle x=1$. Prove that $\displaystyle S$ is a subring of $\displaystyle \mathcal{Z}[x]$.

Do it by definition.
For example, for additive closure say $\displaystyle f(x),g(x) \in S$.
Then $\displaystyle f(x)+g(x) \in S$ since $\displaystyle f(0)+g(0) = 0 + 0 = 0$ and $\displaystyle f(1)+g(1) = 0 + 0 = 0$.